{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true,
    "ExecuteTime": {
     "end_time": "2023-05-11T06:24:39.136218200Z",
     "start_time": "2023-05-11T06:24:11.278526700Z"
    }
   },
   "outputs": [],
   "source": [
    "import torch\n",
    "from torch import nn\n",
    "from d2l import torch as d2l\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "outputs": [
    {
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "T = 1000\n",
    "time = torch.arange(1, T + 1, dtype=torch.float32)\n",
    "x = torch.sin(0.01 * time) + torch.normal(0, 0.2, (T,))\n",
    "d2l.plot(time, x, 'time', 'x', xlim=[1, 1000], figsize=(6, 3))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T06:27:33.906675100Z",
     "start_time": "2023-05-11T06:27:33.812683800Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "接下来，我们将这个序列转换为模型的特征－标签（feature-label）对。基于嵌⼊维度τ，我们将数据映射为数据对yt = xt 和xt = [xt−τ , . . . , xt−1]。这⽐我们提供的数据样本少了τ个，因为我们没有⾜够的历史记录来描述前τ个数据样本。⼀个简单的解决办法是：如果拥有⾜够⻓的序列就丢弃这⼏项；另⼀个⽅法是⽤零填充序列。在这⾥，我们仅使⽤前600个“特征－标签”对进⾏训练。"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "outputs": [],
   "source": [
    "tau = 4\n",
    "features = torch.zeros((T - tau, tau))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T06:31:41.012134900Z",
     "start_time": "2023-05-11T06:31:41.007135800Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "outputs": [],
   "source": [
    "for i in range(tau):\n",
    "    features[:, i] = x[i:T - tau + i]\n",
    "labels = x[tau:].reshape((-1, 1))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T06:38:44.042006800Z",
     "start_time": "2023-05-11T06:38:44.030008900Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "outputs": [
    {
     "data": {
      "text/plain": "torch.Size([996, 1])"
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "labels.shape"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T06:42:53.121035600Z",
     "start_time": "2023-05-11T06:42:53.110035800Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "outputs": [],
   "source": [
    "batch_size, n_train = 16, 600\n",
    "# 只有前n_train个样本用于训练\n",
    "train_iter = d2l.load_array((features[:n_train], labels[:n_train]), batch_size, is_train=True)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T06:43:33.621384200Z",
     "start_time": "2023-05-11T06:43:33.614386800Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "outputs": [],
   "source": [
    "# 初始化⽹络权重的函数\n",
    "def init_weights(m):\n",
    "    if type(m) == nn.Linear:\n",
    "        nn.init.xavier_normal(m.weight)\n",
    "\n",
    "\n",
    "# ⼀个简单的多层感知机\n",
    "def get_net():\n",
    "    net = nn.Sequential(\n",
    "        nn.Linear(4, 10),\n",
    "        nn.ReLU(),\n",
    "        nn.Linear(10, 1)\n",
    "    )\n",
    "    net.apply(init_weights)\n",
    "    return net\n",
    "\n",
    "\n",
    "# 平⽅损失。注意：MSELoss计算平⽅误差时不带系数1/2\n",
    "loss = nn.MSELoss(reduction='none')"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T06:46:33.353908900Z",
     "start_time": "2023-05-11T06:46:33.332891800Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss: 0.066312\n",
      "epoch 2, loss: 0.059041\n",
      "epoch 3, loss: 0.056053\n",
      "epoch 4, loss: 0.055428\n",
      "epoch 5, loss: 0.053466\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\boss\\AppData\\Local\\Temp\\ipykernel_15712\\2449584725.py:4: UserWarning: nn.init.xavier_normal is now deprecated in favor of nn.init.xavier_normal_.\n",
      "  nn.init.xavier_normal(m.weight)\n"
     ]
    }
   ],
   "source": [
    "def train(net, train_iter, loss, epochs, lr):\n",
    "    trainer = torch.optim.Adam(net.parameters(), lr)\n",
    "    for epoch in range(epochs):\n",
    "        for X, y in train_iter:\n",
    "            trainer.zero_grad()\n",
    "            l = loss(net(X), y)\n",
    "            l.sum().backward()\n",
    "            trainer.step()\n",
    "        print(f'epoch {epoch + 1}, '\n",
    "              f'loss: {d2l.evaluate_loss(net, train_iter, loss):f}')\n",
    "\n",
    "\n",
    "net = get_net()\n",
    "train(net, train_iter, loss, 5, 0.01)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T06:51:16.332186100Z",
     "start_time": "2023-05-11T06:51:16.142004800Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "由于训练损失很小，因此我们期望模型能有很好的工作效果。让我们看看这在实践中意味着什么。首先是检查模型预测下一个时间步的能力，也就是单步预测（one-step-ahead prediction）。"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "outputs": [
    {
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124.214992 46.935396 \nL 124.550127 40.359703 \nL 124.885262 41.156624 \nL 125.220397 41.555018 \nL 125.555532 44.767098 \nL 126.225802 55.858237 \nL 126.560938 48.829826 \nL 126.896073 50.599074 \nL 127.231208 50.058345 \nL 127.566343 51.160418 \nL 127.901478 53.57207 \nL 128.236613 51.501849 \nL 128.571748 50.408255 \nL 128.906883 46.816607 \nL 129.242019 52.8506 \nL 129.577154 44.936975 \nL 129.912289 53.889904 \nL 130.247424 51.16389 \nL 130.582559 52.85444 \nL 130.917694 51.959511 \nL 131.252829 57.060405 \nL 131.587965 54.140174 \nL 131.9231 55.532252 \nL 132.258235 50.539323 \nL 132.59337 54.392486 \nL 132.928505 51.781248 \nL 133.26364 57.646709 \nL 133.598775 57.151916 \nL 133.93391 54.424304 \nL 134.269046 59.011256 \nL 134.604181 51.584364 \nL 134.939316 54.541955 \nL 135.274451 50.633548 \nL 135.609586 56.651082 \nL 135.944721 58.024598 \nL 136.279856 57.227918 \nL 136.614992 56.978159 \nL 136.950127 58.970062 \nL 137.285262 57.55572 \nL 137.620397 65.084122 \nL 137.955532 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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "onestep_preds = net(features)\n",
    "d2l.plot([time, time[tau:]],\n",
    "         [x.detach().numpy(), onestep_preds.detach().numpy()], 'time',\n",
    "         'x', legend=['data', '1-step preds'], xlim=[1, 1000],\n",
    "         figsize=(6, 3))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T06:54:54.648094400Z",
     "start_time": "2023-05-11T06:54:54.519094700Z"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "通常，对于直到xt的观测序列，其在时间步t + k处的预测输出xˆt+k 称为k步预测（k-step-ahead-prediction）。由于我们的观察已经到了x604，它的k步预测是xˆ604+k。换句话说，我们必须使用我们自己的预测（而不是原始数据）来进行多步预测。让我们看看效果如何。"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "outputs": [],
   "source": [
    "multistep_preds = torch.zeros(T)\n",
    "multistep_preds[:n_train + tau] = x[:n_train + tau]\n",
    "for i in range(n_train + tau, T):\n",
    "    multistep_preds[i] = net(\n",
    "        multistep_preds[i - tau:i].reshape((1, -1))\n",
    "    )"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T07:01:41.377498600Z",
     "start_time": "2023-05-11T07:01:41.339481300Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "outputs": [
    {
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "d2l.plot([time, time[tau:], time[n_train + tau:]],\n",
    "         [x.detach().numpy(), onestep_preds.detach().numpy(),\n",
    "          multistep_preds[n_train + tau:].detach().numpy()], 'time',\n",
    "         'x', legend=['data', '1-step preds', 'multistep preds'],\n",
    "         xlim=[1, 1000], figsize=(6, 3))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T07:02:35.818302100Z",
     "start_time": "2023-05-11T07:02:35.702299400Z"
    }
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  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "如上⾯的例⼦所⽰，绿线的预测显然并不理想。经过⼏个预测步骤之后，预测的结果很快就会衰减到⼀个常数。为什么这个算法效果这么差呢？事实是由于错误的累积：假设在步骤1之后，我们积累了⼀些错误ϵ1 = ¯ϵ。\n",
    "于是，步骤2的输⼊被扰动了ϵ1，结果积累的误差是依照次序的ϵ2 = ¯ϵ + cϵ1，其中c为某个常数，后⾯的预测误差依此类推。因此误差可能会相当快地偏离真实的观测结果。例如，未来24⼩时的天⽓预报往往相当准确，但超过这⼀点，精度就会迅速下降。\n",
    "\"\"\""
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "outputs": [],
   "source": [
    "max_steps = 64"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T07:06:52.586358600Z",
     "start_time": "2023-05-11T07:06:52.573356400Z"
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  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "outputs": [],
   "source": [
    "features = torch.zeros((T - tau - max_steps + 1, tau + max_steps))\n",
    "# 列i（i<tau）是来⾃x的观测，其时间步从（i）到（i+T-tau-max_steps+1）\n",
    "for i in range(tau):\n",
    "    features[:, i] = x[i: i + T - tau - max_steps + 1]\n",
    "\n",
    "# 列i（i>=tau）是来⾃（i-tau+1）步的预测，其时间步从（i）到（i+T-tau-max_steps+1）\n",
    "for i in range(tau, tau + max_steps):\n",
    "    features[:, i] = net(features[:, i - tau:i]).reshape(-1)"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T07:11:45.845922200Z",
     "start_time": "2023-05-11T07:11:45.826922Z"
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  {
   "cell_type": "code",
   "execution_count": 23,
   "outputs": [
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89.134364 \nL 151.759731 97.691997 \nL 152.096214 81.273685 \nL 152.432696 84.695465 \nL 152.769179 73.08868 \nL 153.105661 86.554285 \nL 153.442144 72.278158 \nL 153.778626 89.727182 \nL 154.115108 84.61738 \nL 154.451591 91.482196 \nL 154.788073 93.926722 \nL 155.124556 75.511206 \nL 155.461038 86.444934 \nL 155.79752 77.11467 \nL 156.134003 93.280579 \nL 156.470485 94.250065 \nL 156.806968 91.060766 \nL 157.14345 79.212539 \nL 157.479932 94.286307 \nL 157.816415 72.403904 \nL 158.152897 90.250743 \nL 158.48938 92.186115 \nL 158.825862 88.889327 \nL 159.162345 98.583646 \nL 159.498827 94.311025 \nL 159.835309 79.513671 \nL 160.171792 102.878283 \nL 160.508274 100.341072 \nL 160.844757 94.268021 \nL 161.181239 100.392323 \nL 161.517721 92.012863 \nL 161.854204 115.329829 \nL 162.190686 118.391898 \nL 162.527169 111.471375 \nL 162.863651 114.517369 \nL 163.200133 113.487788 \nL 163.536616 99.20308 \nL 163.873098 112.10472 \nL 164.209581 113.192568 \nL 164.546063 117.209431 \nL 164.882546 106.315786 \nL 165.219028 112.5016 \nL 165.55551 114.063794 \nL 165.891993 99.920503 \nL 166.564958 124.16803 \nL 166.90144 116.668531 \nL 167.237922 105.831642 \nL 167.574405 121.14669 \nL 167.910887 119.315247 \nL 168.24737 126.70174 \nL 168.583852 123.138916 \nL 168.920334 111.418799 \nL 169.256817 115.206668 \nL 169.593299 120.306719 \nL 169.929782 118.780859 \nL 170.266264 112.98479 \nL 170.602747 111.769851 \nL 170.939229 122.405001 \nL 171.275711 122.959419 \nL 171.612194 118.677273 \nL 171.948676 124.975934 \nL 172.285159 124.828252 \nL 172.621641 120.507057 \nL 172.958123 122.366977 \nL 173.294606 130.643066 \nL 173.631088 122.504659 \nL 173.967571 119.845623 \nL 174.304053 123.011769 \nL 174.640535 131.011912 \nL 174.977018 124.598099 \nL 175.3135 121.243175 \nL 175.986465 142.653925 \nL 176.322948 138.543458 \nL 176.65943 126.788765 \nL 176.995912 137.096706 \nL 177.332395 133.231348 \nL 177.668877 126.817885 \nL 178.00536 137.950822 \nL 178.341842 133.345854 \nL 178.678324 135.481466 \nL 179.351289 138.103856 \nL 179.687772 132.913144 \nL 180.024254 125.281777 \nL 180.360736 128.26358 \nL 180.697219 127.473694 \nL 181.033701 122.006732 \nL 182.379631 138.533396 \nL 182.716113 134.028375 \nL 183.052596 138.382996 \nL 183.725561 136.415149 \nL 184.062043 142.952597 \nL 184.398525 140.99694 \nL 184.735008 144.20272 \nL 185.07149 133.871549 \nL 185.407973 132.151018 \nL 185.744455 141.977457 \nL 186.080938 133.651006 \nL 186.41742 149.770778 \nL 186.753902 142.576635 \nL 187.090385 142.49793 \nL 187.426867 138.485102 \nL 187.76335 140.792433 \nL 188.099832 147.343618 \nL 188.436314 135.50312 \nL 188.772797 138.478257 \nL 189.109279 147.401321 \nL 189.445762 128.280183 \nL 189.782244 132.00702 \nL 190.118726 132.318247 \nL 190.455209 140.999605 \nL 190.791691 146.676326 \nL 191.128174 138.602832 \nL 191.464656 137.944451 \nL 191.801139 139.792866 \nL 192.137621 127.097499 \nL 192.474103 135.35115 \nL 192.810586 139.81229 \nL 193.147068 132.900744 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151.026223 \nL 207.279329 153.069534 \nL 207.615812 144.983244 \nL 207.952294 149.852496 \nL 208.288777 145.255516 \nL 208.625259 145.654413 \nL 208.961742 149.458128 \nL 209.298224 145.263239 \nL 209.634706 158.993952 \nL 210.307671 142.539396 \nL 210.644154 142.468476 \nL 210.980636 144.248521 \nL 211.317118 157.288564 \nL 211.653601 142.890737 \nL 212.326566 150.760983 \nL 212.663048 145.303267 \nL 213.336013 151.018523 \nL 213.672495 155.593167 \nL 214.008978 156.936819 \nL 214.34546 150.06081 \nL 214.681943 140.092152 \nL 215.018425 147.370702 \nL 215.354907 145.750583 \nL 215.69139 147.873492 \nL 216.027872 146.425428 \nL 216.364355 138.985717 \nL 216.700837 146.38296 \nL 217.037319 147.570477 \nL 217.373802 146.570863 \nL 217.710284 143.77844 \nL 218.046767 143.554548 \nL 218.383249 138.394906 \nL 218.719731 150.216098 \nL 219.056214 136.90376 \nL 219.392696 141.663479 \nL 219.729179 148.540435 \nL 220.065661 152.546986 \nL 220.402144 143.690249 \nL 220.738626 145.322892 \nL 221.075108 141.120979 \nL 221.411591 142.768609 \nL 221.748073 146.776551 \nL 222.084556 148.304214 \nL 222.421038 141.767162 \nL 222.75752 146.180714 \nL 223.094003 142.330313 \nL 223.430485 133.449024 \nL 223.766968 142.917852 \nL 224.10345 134.128064 \nL 224.776415 152.89459 \nL 225.112897 141.247442 \nL 225.44938 142.605692 \nL 225.785862 141.623769 \nL 226.122345 143.994041 \nL 226.458827 138.804627 \nL 226.795309 146.480124 \nL 227.131792 144.285597 \nL 227.468274 137.41 \nL 227.804757 148.714276 \nL 228.477721 139.046396 \nL 228.814204 151.449082 \nL 229.150686 146.296844 \nL 229.487169 137.826837 \nL 229.823651 138.998078 \nL 230.160133 150.011202 \nL 230.496616 136.159473 \nL 230.833098 135.662944 \nL 231.169581 139.613113 \nL 231.506063 140.48715 \nL 231.842546 134.245227 \nL 232.179028 135.799414 \nL 232.51551 139.599648 \nL 232.851993 135.453924 \nL 233.188475 136.10309 \nL 233.524958 141.428578 \nL 233.86144 143.961083 \nL 234.534405 136.100503 \nL 234.870887 151.554303 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99.400785 \nL 261.78948 91.261169 \nL 262.125963 98.463138 \nL 262.462445 99.115185 \nL 263.13541 88.05451 \nL 263.471892 76.547219 \nL 263.808375 84.473528 \nL 264.144857 80.447479 \nL 264.48134 87.08265 \nL 264.817822 83.88166 \nL 265.154304 77.048835 \nL 265.490787 77.445921 \nL 265.827269 86.580977 \nL 266.163752 65.667957 \nL 266.500234 78.713841 \nL 266.836716 58.229388 \nL 267.173199 56.10355 \nL 267.509681 72.485858 \nL 267.846164 56.2229 \nL 268.182646 72.117432 \nL 268.519128 57.734676 \nL 269.192093 90.181226 \nL 269.865058 65.155733 \nL 270.201541 60.123323 \nL 270.538023 50.443743 \nL 270.874505 51.696143 \nL 271.210988 62.489854 \nL 271.54747 63.506078 \nL 271.883953 87.26614 \nL 272.220435 83.215402 \nL 272.556917 82.991067 \nL 272.8934 77.63686 \nL 273.229882 63.315644 \nL 273.566365 71.972789 \nL 273.902847 67.743574 \nL 274.239329 78.438344 \nL 274.575812 70.809263 \nL 274.912294 76.577239 \nL 275.248777 49.51083 \nL 275.585259 45.25718 \nL 275.921742 48.250016 \nL 276.258224 57.512461 \nL 276.594706 56.527141 \nL 276.931189 72.565937 \nL 277.267671 54.155507 \nL 277.604154 76.778379 \nL 278.277118 51.339185 \nL 278.613601 56.57636 \nL 278.950083 50.642889 \nL 279.286566 49.406157 \nL 279.623048 66.7269 \nL 279.95953 55.360785 \nL 280.296013 57.355728 \nL 280.968978 43.383926 \nL 281.30546 42.717162 \nL 281.641943 42.672783 \nL 281.978425 48.43684 \nL 282.314907 43.968505 \nL 282.65139 41.574637 \nL 282.987872 50.351458 \nL 283.324355 44.665057 \nL 283.660837 59.554958 \nL 283.997319 50.510693 \nL 284.333802 50.013908 \nL 284.670284 42.975892 \nL 285.006767 39.550159 \nL 285.343249 39.307947 \nL 285.679731 39.525965 \nL 286.016214 41.04115 \nL 286.352696 43.157898 \nL 286.689179 39.173661 \nL 287.025661 38.069579 \nL 287.698626 40.962018 \nL 288.035108 38.521868 \nL 288.371591 32.294024 \nL 288.708073 33.951708 \nL 289.044556 30.194998 \nL 289.381038 41.063192 \nL 289.71752 35.51044 \nL 290.054003 40.280508 \nL 290.390485 40.527506 \nL 290.726968 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320.33742 32.95559 \nL 320.673902 32.621373 \nL 321.010385 33.485776 \nL 321.346867 30.130333 \nL 321.68335 30.703887 \nL 322.019832 29.572908 \nL 322.692797 34.715201 \nL 323.029279 33.430349 \nL 323.365762 37.247714 \nL 324.038726 30.836069 \nL 324.711691 30.619014 \nL 325.048174 34.380603 \nL 325.384656 34.002807 \nL 325.721139 31.666837 \nL 326.057621 31.242363 \nL 326.394103 28.765044 \nL 326.730586 32.174142 \nL 327.067068 34.240886 \nL 327.403551 38.871298 \nL 327.740033 38.409142 \nL 328.076515 35.493115 \nL 328.412998 35.06923 \nL 328.74948 36.085471 \nL 329.085963 34.237296 \nL 329.422445 36.347028 \nL 329.758927 34.782919 \nL 330.09541 35.811638 \nL 330.768375 34.620554 \nL 331.104857 31.298769 \nL 331.44134 33.108018 \nL 331.777822 36.849081 \nL 332.114304 38.460731 \nL 332.450787 38.494629 \nL 333.123752 31.311192 \nL 333.460234 34.880775 \nL 333.796716 33.601284 \nL 334.133199 35.69066 \nL 334.469681 34.28615 \nL 334.806164 35.754268 \nL 335.479128 40.247535 \nL 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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "steps = (1, 4, 16, 64)\n",
    "d2l.plot([time[tau + i - 1: T - max_steps + i] for i in steps],\n",
    "         [features[:, tau + i - 1].detach().numpy() for i in steps], 'time', 'x',\n",
    "         legend=[f'{i}-step preds' for i in steps], xlim=[5, 1000],\n",
    "         figsize=(6, 3))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2023-05-11T07:11:54.138827500Z",
     "start_time": "2023-05-11T07:11:54.017825Z"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [],
   "metadata": {
    "collapsed": false
   }
  }
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